my_array.concat('foo');

concat doesn't alter the original array.

Use the Array.prototype.push method to append values to the end of an array:

// initialize array
var arr = [
  "Hi",
  "Hello",
  "Bonjour"
];

// append new value to the array
arr.push("Hola");

console.log(arr);

You can use the push() function to append more than one value to an array in a single call:

// initialize array
var arr = ["Hi", "Hello", "Bonjour", "Hola"];

// append multiple values to the array
arr.push("Salut", "Hey");

// display all values
for (var i = 0; i < arr.length; i++) {
  console.log(arr[i]);
}

Note that the push() method returns the updated length of the array.

Update

If you want to add the items of one array to another array, you can use firstArray.concat(secondArray):

var arr = [
  "apple",
  "banana",
  "cherry"
];

// Do not forget to assign the result as, unlike push, concat does not change the existing array
arr = arr.concat([
  "dragonfruit",
  "elderberry",
  "fig"
]);

console.log(arr);

Update

Just an addition to this answer if you want to prepend any value to the start of an array (i.e. first index) then you can use Array.prototype.unshift for this purpose.

var arr = [1, 2, 3];
arr.unshift(0);
console.log(arr);

It also supports appending multiple values at once just like push.

Update

Another way with ES6 syntax is to return a new array with the spread syntax. This leaves the original array unchanged, but returns a new array with new items appended or prepended, compliant with the spirit of functional programming.

const arr1 = [
  "Hi",
  "Hello",
  "Bonjour",
];
const arr2 = [
  "Ciao",
  "Hej",
  "Merhaba",
];

const newArr1 = [
  ...arr1,
  "Salut",
];
const newArr2 = [
  "Salut",
  ...arr2,
];
const newArr3 = [
  ...arr1,
  ...arr2,
];

console.log(newArr1, newArr2, newArr3);

The .push method can take multiple arguments. You can use the spread operator to pass all the elements of the second array as arguments to .push:

>>> a.push(...b)

If your browser does not support ECMAScript 6, you can use .apply instead:

>>> a.push.apply(a, b)

Or perhaps, if you think it's clearer:

>>> Array.prototype.push.apply(a,b)

Please note that all these solutions will fail with a stack overflow error if array b is too long (trouble starts at about 100,000 elements, depending on the browser).
If you cannot guarantee that b is short enough, you should use a standard loop-based technique described in the other answer.